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Harvard University Biostatistics Working Paper Series
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A unification of mediation and interaction: a
four-way decomposition
Tyler J. VanderWeele
∗∗Harvard University, [email protected]
This working paper is hosted by The Berkeley Electronic Press (bepress) and may not be commer-cially reproduced without the permission of the copyright holder.
http://biostats.bepress.com/harvardbiostat/paper170 Copyright c2014 by the author.
A unification of mediation and interaction: a
four-way decomposition
Tyler J. VanderWeele
Abstract
It is shown that the overall effect of an exposure on an outcome, in the pres-ence of a mediator with which the exposure may interact, can be decomposed into four components: (i) the effect of the exposure in the absence of the mediator, (ii) the interactive effect when the mediator is left to what it would be in the ab-sence of exposure, (iii) a mediated interaction, and (iv) a pure mediated effect. These four components, respectively, correspond to the portion of the effect that is due to neither mediation nor interaction, to just interaction (but not mediation), to both mediation and interaction, and to just mediation (but not interaction). This four-way decomposition unites methods that attribute effects to interactions and methods that assess mediation. Certain combinations of these four components correspond to measures for mediation, while other combinations correspond to measures of interaction previously proposed in the literature. Prior decomposi-tions in the literature are in essence special cases of this four-way decomposition. The four-way decomposition can be carried out using standard statistical models, and software is provided to estimate each of the four components. The four-way decomposition provides maximum insight into how much of an effect is mediated, how much is due to interaction, how much is due to both mediation and interaction together, and how much is due to neither.
Introduction
Methodology for mediation and interaction has developed rapidly over the past decade. Methods for e¤ect decomposition to assess direct and indirect e¤ects have shed light on mechanisms and pathways.1 19 Other methods and measures have been useful in assessing how much of the e¤ect of one exposure is due to its interaction with another.20 24 In this paper we provide theory and methods to unite these e¤ect decomposition and attribution methods for mediation and interaction. The paper’s central result is that the total e¤ect of an exposure on an outcome, in the presence of a mediator with which the exposure may interact, can be decomposed into four components: components due to just mediation, to just interaction, to both mediation and interaction, and to neither mediation nor interaction.
After presenting this four-way decomposition, we will discuss assumptions for identifying these four components from data and we will relate this four-way decomposition approach to various statistical models. We then discuss the relations between existing measures of mediation and interaction and each of the four components, and show how existing measures of mediation and interaction consist of di¤erent combinations of these four components. We show how di¤erent e¤ect decomposition and attribution approaches for mediation and interaction can in fact be united within this four-fold framework; when some of the components are combined, the framework presented in this paper essentially collapses to approaches that have been used previously. The greatest insight, however, is arguably gained when the four-fold approach is employed and we illustrate this with an example from genetic epidemiology.
Notation
Let A denote the exposure of interest, Y the outcome, and M a potential mediator, and let C denote a set of baseline covariates. We will suppose we want to compare two levels of the exposure, aand a ; for binary exposure we would have a= 1 and a = 0. For simplicity we will consider the setting of a binary exposure and binary mediator; however more general results that are applicable to arbitrary exposures and mediators are given in the Appendix. We let Ya and Ma
denote respectively the potentially counterfactual values of the outcome and mediator that would have been observed had the exposure Abeen set to level a. The total e¤ect (T E) of the exposure A on the outcomeY is de…ned asY1 Y0; the total e¤ect of the exposureA on the mediatorM is
de…ned asM1 M0. We will not in general ever know what these e¤ects are at the individual level
but we might hope to be able to estimate them on average for a population. For the …rst part of this paper, however, we will be concerned with concepts and only later will we turn to what can be identi…ed with data and under what assumptions.
We will also need counterfactuals of another form. Let Yam denote the value of the outcome
that would have been observed hadAbeen set to levela, andM tom. The controlled direct e¤ect, comparing exposure levelA= 1toA= 0and …xing the mediator to levelmis de…ned byY1m Y0m
and captures the e¤ect of exposureAon outcomeY, intervening to …xM tom; it may be di¤erent for di¤erent levels of m.1;2 It may also be di¤erent for persons. Finally we will also later consider counterfactuals of the formYaMa which is the outcomeY that would have occurred if we …xed A
to aand we …xed M to the level it would have taken if A had beena . We will also make some technical assumptions referred to as consistency and composition that are also needed to relate the observed data to counterfactual quantities. The consistency assumption in this context is that when A = a, the counterfactual outcomes Ya and Ma are equal to the observed outcomes Y and
M, respectively, and that whenA=aandM =m, the counterfactual outcomeYamis equal toY.
A Four-Fold Decomposition
We show in the Appendix that we can decompose the total e¤ect (T E) of A on Y into the following four components:
Y1 Y0 = (Y10 Y00) + (Y11 Y10 Y01+Y00)(M0) (1)
+(Y11 Y10 Y01+Y00)(M1 M0) + (Y01 Y00)(M1 M0):
The …rst component,(Y10 Y00), is the direct e¤ect of the exposureAif the mediator were removed,
i.e. …xed toM = 0. This e¤ect is sometimes referred to as a ’controlled direct e¤ect’(CDE).1;2 The second component,(Y11 Y10 Y01+Y00)(M0), we will call a ’reference interaction’(IN Tref). The
term(Y11 Y10 Y01+Y00)is an additive interaction. It can be rewritten as(Y11 Y00) f(Y10 Y00)+ (Y01 Y00)gand will be non-zero for a person if the e¤ect on the outcome of setting both the exposure
and the mediator to present di¤ers from the sum of the e¤ect of having only the exposure present and the e¤ect of having only the mediator present; additive interaction is generally considered of greatest public health importance20 22. The second component in the decomposition in (1) is the product of this additive interaction and M0. Thus this second component, (Y11 Y10 Y01+Y00)(M0),
is an additive interaction that only operates if the mediator is present in the absence of exposure i.e. whenM0 = 1. The third component, (Y11 Y10 Y01+Y00)(M1 M0), will be referred to as
a ’mediated interaction’(IN Tmed). It is the same additive interaction contrast times(M1 M0).
In other words it is an additive interaction that only operates if the exposure has an e¤ect on the mediator so that M1 M0 6= 0. The …nal component, (Y01 Y00)(M1 M0), is the e¤ect of the
mediator in the absence of the exposure,Y01 Y00, multiplied by the the e¤ect of the exposure on
the mediator itself, M1 M0. It will be non-zero only if the mediator a¤ects the outcome when
the exposure is absent, and the exposure itself a¤ects the mediator. We might refer to this …nal component as a ’mediated main e¤ect’ or, as will be explained below, as a ’pure indirect e¤ect’ (P IE).1;2
The intuition behind this decomposition is that if the exposure a¤ects the outcome for a par-ticular individual, then at least one of four things must be the case. Either the exposure might a¤ect the outcome through pathways which do not require the mediator (i.e. the exposure a¤ects the outcome even when the mediator is absent); in other words the …rst component is non-zero. Or alternatively, the exposure e¤ect might operate only in the presence of the mediator (i.e. there is an interaction) and it might also be the case that the exposure itself is not necessary for the mediator to be present (i.e. the mediator itself would be present in the absence of the exposure, though the mediator is itself necessary for the exposure to have an e¤ect on the outcome); in other words, the second component is non-zero. Or alternatively, the exposure e¤ect might operate only in the presence of the mediator (i.e. there is an interaction) and it might also be the case that the exposure itself is in fact needed for the mediator to be present (i.e. the exposure causes the mediator, and the presence of the mediator is itself necessary for the exposure to have an e¤ect on the outcome); in other words, the third component in non-zero. Or …nally, it might alternatively be the case that the mediator can cause the outcome in the absence of the exposure, but the exposure is necessary for the mediator itself to be present; in other words, the fourth component is non-zero. The decomposition above, proved in the Appendix, provides a mathematical formalization of this intuition. We could thus rewrite our decomposition as:
T E=CDE+IN Tref+IN Tmed+P IE:
to know the value of each of the four components for a particular individual, but below we will discuss assumptions under which we could estimate measures of these four components on average for a particular population. We will see below that under certain assumptions about confounding the average value of each of four components is given by the following empirical expressions:
E[CDE] = (p10 p00)
E[IN Tref] = (p11 p10 p01+p00)P(M = 1jA= 0)
E[IN Tmed] = (p11 p10 p01+p00)fP(M = 1jA= 1) P(M = 1jA= 0)g
E[P IE] = (p01 p00)fP(M = 1jA= 1) P(M = 1jA= 0)g:
where pam = E(YjA = a; M = m). If we let pa =E(YjA = a) we will have following empirical
decomposition:
pa=1 pa=0 = (p10 p00) + (p11 p10 p01+p00)P(M = 1jA= 0)
+ (p11 p10 p01+p00)fP(M = 1jA= 1) P(M = 1jA= 0)g
+ (p01 p00)fP(M = 1jA= 1) P(M = 1jA= 0)g (1b)
With such average measures we would be able to assess how much of the total e¤ect is due to (i) neither mediation nor interaction (the …rst component); how much is due to interaction but not mediation (the second component), how much is due to both mediation and interaction (the third component); and how much of the e¤ect is due to mediation but not interaction (the fourth component). The four components of the total e¤ect are summarized in Table 1.
Table 1. The Four Basic Components of the Total E¤ect (the following four components sum to the total e¤ectT E =Y1 Y0)
E ¤e c t C o u nt e rfa c t u a l D e …n itio n E m p iric a l A n a lo g u e C o nt ro lle d D ire c t E ¤e c t(CDE) (Y10 Y00) (p10 p00)
R e fe re n c e Inte ra c tio n(IN Tref) (Y11 Y10 Y01+Y00)(M0) (p11 p10 p01+p00)P(M= 1jA= 0)
M e d ia te d Int e ra c tio n(IN Tmed) (Y11 Y10 Y01+Y00)(M1 M0) (p11 p10 p01+p00)fP(M= 1jA= 1) P(M= 1jA= 0)g
P u re In d ire c t E ¤e c t(P IE) (Y01 Y00)(M1 M0) = (Y0M1 Y0M0) (p01 p00)fP(M= 1jA= 1) P(M= 1jA= 0)g:
If we let E[T E] denote the average total e¤ect for the population (equal to pa=1 pa=0 =
E(YjA= 1) E(YjA= 0)in the absence of confounding), then we could also consider the propor-tion of the total e¤ect that is due to each of these four components using the ratios EE[CDE[T E]], E[IN Tref]
E[T E] ,
E[IN Tmed]
E[T E] , and
E[P IE]
E[T E]. We could also assess the overall proportion due to mediation by summing the
proportions due to the mediated interaction and to the pure indirect e¤ect, i.e. E[IN Tmed]+E[P IE]
E[T E] .
We could likewise assess the overall proportion due to interaction by summing the proportions due to the reference interaction and to the mediated interaction, i.e. E[IN Tref]+E[IN Tmed]
E[T E] . Such
proportion measures, however, generally only make sense reporting if all of the components are in the same direction (e.g. all positive or all negative). The statistical properties of such proportion measures can also be highly variable, and hence problematic, if the total e¤ect is close to zero as might be the case if some of the components were positive and others negative. Similar comments pertain to other proportion measures described below.
We will …rst consider the no-confounding assumptions that allow us to estimate these four components on average, and statistical methods to carry out such estimation. We will later consider the relationships between this four-fold decomposition and other concepts from the literatures on mediation and interaction that involve e¤ect decomposition and attribution.
Identi…cation of the E¤ects
Our discussion thus far has been primarily conceptual. As we have noted, the individual level e¤ects in the four-way decomposition cannot be identi…ed from the data, but under certain no-confounding assumptions the four components can be identi…ed from the data on average for a pop-ulation. As discussed further in the Appendix, for a causal diagram interpreted as non-parametric structural equation models of Pearl,18 the following four assumptions su¢ ce to identify each of the
four components from the data: (i) the e¤ect the exposureA on the outcomeY is unconfounded conditional on C; (ii) the e¤ect the mediatorM on the outcome Y is unconfounded conditional on
(C; A); (iii) the e¤ect the exposure A on the mediatorM is unconfounded conditional on C; and (iv) none of the mediator-outcome confounders are themselves a¤ected by the exposure. These are the same four assumptions that are often used in the literature on mediation.2;4;5 If we letX??YjZ denote thatXis independent ofY conditional onZ, then these four assumptions stated formally in terms of counterfactual independence are: (i)Yam??AjC, (ii)Yam??MjfA; Cg, (iii)Ma??AjC,
and (iv) Yam ?? Ma jC. Note that assumption (iv) requires that none of the mediator-outcome
confounders are themselves a¤ected by the exposure. This assumption would hold in Figure 1 but would be violated in Figure 2.
Figure 1: Mediation with exposure A, outcomeY, mediatorM, and confounders C.
Figure 2: Mediation with a mediator-outcome confounder Lthat is a¤ected by the exposure.
above:
E[CDE] = (p10 p00)
E[IN Tref] = (p11 p10 p01+p00)P(M = 1jA= 0)
E[IN Tmed] = (p11 p10 p01+p00)fP(M = 1jA= 1) P(M = 1jA= 0)g
E[P IE] = (p01 p00)fP(M = 1jA= 1) P(M = 1jA= 0)g:
More general formulae involving covariates and with arbitrary exposures and mediator (rather than binary) are given in the Appendix.
The counterfactual statement of assumption (iv), Yam ?? Ma jC, is somewhat controversial
as it involves what are sometimes called ’cross-world’ independencies. It would hold in Figure 1 interpreted as a non-parametric structural equation model,18 but may not hold under other interpretations of causal diagrams.19 We noted above that the empirical equivalent of our
four-way decomposition was (1b). As shown in the Appendix, this decomposition holds without any assumptions at all about confounding. However, to interpret each of the components causally does require assumptions about confounding. Assumptions (i)-(iv) above allow for interpreting each of the components as population average causal e¤ects of each of the four components in the four-way individual level counterfactual decomposition: CDE,IN Tref,IN Tmed, andP IE. In the Appendix
we also discuss how a slightly weaker interpretation is also valid essentially under just assumptions (i)-(iii) alone, without requiring the more controversial assumptions (iv).
Also of interest is the fact that the controlled direct e¤ect, CDE, only requires assumption (i) and (ii) to be identi…ed.1;2 This does not require the more controversial cross-world independence
assumptions. The average controlled direct e¤ect is sometimes subtracted from the average total e¤ect to get a proportion eliminated measureE[P E] :=E[T E] E[CDE]. Whenever we can iden-tify the total e¤ect and the controlled direct e¤ect we can calculate this portion eliminated measure. Interestingly, as described further below, the four-way decomposition gives a more mechanistic in-terpretation of this portion eliminated measure: the portion eliminated is the sum of the reference interaction, the mediated interaction, and the pure indirect e¤ect (P E=IN Tref+IN Tmed+P IE)
i.e. it is the portion due to either mediation or interaction or both. We cannot empirically separate apart these three components without using stronger assumptions such as (i)-(iv) above. However, whenever we can identify the total e¤ect and the controlled direct e¤ect (which we can do under much weaker assumptions) we can obtain also the sum of the three other components since they are simply the di¤erence between the total e¤ect and the controlled direct e¤ect.
Relation to Statistical Models
Suppose that assumptions (i)-(iv) hold, that Y and M are continuous and that the following regression models for Y and M are correctly speci…ed:
E[Yja; m; c] = 0+ 1a+ 2m+ 3am+ 04c
E[Mja; c] = 0+ 1a+ 02c:
It is shown in the eAppendix that for exposure levels a and a , and for setting the mediator to 0
the four components are given by:
E[CDEjc] = 1(a a )
E[IN Trefjc] = 3( 0+ 1a + 02c)(a a )
E[IN Tmedjc] = 3 1(a a )(a a )
E[P IEjc] = ( 2 1+ 3 1a )(a a )
If the exposure were binary, the pure direct, reference interaction, mediated interaction, and pure indirect e¤ects would, respectively, simply be: 1, 3( 0+ 20c)g, 3 1, and 2 1. Standard errors
for estimators of these quantities could be derived using the delta method along the lines of Van-derWeele and Vansteelandt4 or by using bootstrapping. SAS code to implement this approach to obtain estimates and con…dence intervals is provided in the eAppendix. The eAppendix likewise provides a straightforward modeling approach, and SAS code, when the mediator is binary rather continuous.
Binary Outcomes and the Ratio Scale
Thus far we have been considering the de…nition of these four components on a di¤erence scale. Often in epidemiology risk ratios or odds ratios are used for convenience, or ease of interpretation, or to account for study design. By dividing the decomposition in (1b) by pa=0 we can rewrite this
decomposition on the ratio scale as
RRa=1 1 = (RR10 1) + (RR11 RR10 RR01+ 1)P(M = 1jA= 0) (2)
+ (RR11 RR10 RR01+ 1)fP(M = 1jA= 1) P(M = 1jA= 0)g
+ (RR01 1)fP(M = 1jA= 1) P(M = 1jA= 0)g
whereRRa=1= ppaa=1=0 is the relative risk for exposureA comparingA= 1 to the reference category
A= 0, andRRam= ppam00 is the relative risk for comparing categoriesA=a; M =mto the reference
category A = 0; M = 0, and where is a scaling factor which is given by = p00
pa=0. Note also here that the term,(RR11 RR10 RR01+ 1), is Rothman’s excess relative risk due to interaction
(RERI), and is a measure of additive interaction using ratios.20
The decomposition in (2) involves decomposing the excess relative risk for the exposure A, RRa=1 1, into four components on the excess relative risk scale involving, as before, (i) the
controlled direct e¤ect ofA when M = 0, (ii) a reference interaction, (iii) a mediated interaction, and (iv) a mediated main e¤ect. Note that although the right hand side of the decomposition involves a scaling factor , if what we are interested in is the proportion of the e¤ect attributable to each of the components, then if we take any particular component and divide it by the sum of all the components, then the scaling drops out. The proportion of the e¤ect attributable to each of the four components is thus given by the expressions in Table 2.
Table 2. Proportion attributable to the controlled direct e¤ect(P ACDE), the reference interaction
(P AIN T ref), the mediated interaction (P AIN T med) and the pure indirect e¤ect (P AP IE) when
using a ratio scale.
P ACDE = (RR10 1) (RR10 1) + (RERI)P(M = 1jA= 1) + (RR01 1)fP(M= 1jA= 1) P(M= 1jA= 0)g P AIN T ref = (RERI)P(M= 1jA= 0) (RR10 1) + (RERI)P(M = 1jA= 1) + (RR01 1)fP(M= 1jA= 1) P(M= 1jA= 0)g P AIN T med = (RERI)fP(M= 1jA= 1) P(M = 1jA= 0)g (RR10 1) + (RERI)P(M = 1jA= 1) + (RR01 1)fP(M= 1jA= 1) P(M= 1jA= 0)g P AP IE = (RR01 1)fP(M= 1jA= 1) P(M= 1jA= 0)g (RR10 1) + (RERI)P(M = 1jA= 1) + (RR01 1)fP(M= 1jA= 1) P(M= 1jA= 0)g :
The four-fold proportion attributable measures given in Table 2 allow us to estimate the propor-tion of the total e¤ect attributable just to mediapropor-tion(P AP IE), just due to interaction(P AIN T ref),
due to both mediation and interaction (P AIN T med), or due to neither mediation nor interaction
(P ACDE). Further technical details concerning the four-way decomposition on the ratio scale and
for obtaining estimates and con…dence intervals using logistic regression for the outcome along with linear regression for a continuous mediator or a second logistic regression for a binary mediator is given in the eAppendix. SAS code to implement this approach is also given in the eAppendix. Illustration
We will consider a data example from genetic epidemiology to illustrate the four-way decompo-sition. Speci…cally, we consider the extent to which the e¤ect of chromosome 15q25.1 rs8034191 C alleles on lung cancer risk is mediated by cigarettes smoked per day and/or due to interaction with this smoking measure. rs8034191 C alleles had been found to be associated with both smoking26;27 and lung cancer28 30but there had been debate as to whether the e¤ects on lung cancer were direct or mediated by smoking. VanderWeele et al.31 used methods from the causal mediation analysis literature to assess whether the e¤ect was direct or indirect and found that most of the e¤ect was not mediated by cigarettes per day (the total indirect e¤ect was very small and the pure direct e¤ect was large). In large meta-analyses, Truong et al.32 found no association between the genetic variants amongst never smokers suggesting strong interaction between the variants and smoking behavior; VanderWeele et al.31 likewise reported statistical evidence of interaction. Here we will use the four-way decomposition to assess how much of the e¤ect is due to each of the components. We use data on 1836 cases and 1452 controls from a lung cancer case-control study at Massa-chusetts General Hospital; see Miller et al.33 or VanderWeele et al.31 for further details on the study. As the exposure we compare2versus0C alleles, and use cigarettes per day as the mediator (the square root of this measure is used so that the measure is more normally distributed). Covari-ates adjusted for in the analysis include sex, age, education, and smoking duration. Analyses are restricted to Caucasians. Because the outcome, lung cancer, is rare, odds ratios approximate risk ratios. We …t a logistic regression model for lung cancer on the variants, smoking, their interac-tion, and the covariates; and a linear regression model for smoking on the variants and covariates. Con…dence intervals are obtained using the delta method. Details of this modeling approach in the context of the four-way decomposition are given in the eAppendix; SAS code is also provided. Results are summarized in Table 3.
Table 3. Proportions of the e¤ect of genetic variants on lung cancer due to mediation and interaction with smoking (cigarettes per day)
Component Excess Relative Risk Proportion Attributable Other Proportions
CDE 0:30(95% CI: 0:19;0:79) 39%(95% CI: 11%;89%) Overall Proportion to Interaction: IN Tref 0:42(95% CI:0:11;0:73) 55%(95% CI:8%;101%) 59%(95% CI:9%;109%)
IN Tmed 0:034(95% CI: 0:02;0:09) 4%(95% CI: 3%;11%) Overall Proportion to Mediation: P IE 0:014(95% CI: 0:01;0:04) 2%(95% CI: 1%;5%) 6%(95% CI: 3%;15%)
Total 0:77(95% CI:0:33;1:21) 100%
The overall risk ratio comparing2 versus0 C alleles was1:77 (95% CI:1:33;2:21) for an excess relative risk of 1:77 1 = 0:77 (95% con…dence interval = 0:33;1:21). We decompose this excess relative risk into the four components. The component due to the pure indirect e¤ect is0:014(95% CI: 0:01;0:04); the component due to the mediated interaction is 0:034 (95% CI: 0:02;0:09); the component due to the reference interaction is 0:42 (95% CI: 0:11;0:73); and the component due to the controlled direct e¤ect (if smoking were …xed to 0) is 0:30 (95% CI: 0:19;0:79). The four components sum to the excess relative risk: 0:014 + 0:034 + 0:42 + 0:30 0:77. Of the four components, the reference interaction is most substantial, highlighting the important role of interaction in this context. The overall proportion mediated (the sum of the pure indirect e¤ect and the mediated interaction, divided by the excess relative risk) is quite small 6:2% (95% CI:
2:7%;15:1%), as had been indicated in the analyses of VanderWeele et al.31The overall proportion
attributable to interaction (the reference interaction plus the mediated interaction, divided by the excess relative risk) is relatively substantial59:2%(95% CI: 9:2%;109:3%). Mediation may play a role here (and probably does as the variants do a¤ect smoking and smoking a¤ects lung cancer) but interaction, between the variants and smoking, is clearly much more important in this context. Relation to Mediation Decompositions
We will …rst discuss the relations between the four components above and concepts from the me-diation analysis literature, and we will then discuss relations with the interaction analysis literature. As above, our four-fold decomposition is:
Y1 Y0 = (Y10 Y00) + (Y11 Y10 Y01+Y00)(M0)
+(Y11 Y10 Y01+Y00)(M1 M0) + (Y01 Y00)(M1 M0):
The …rst component,(Y10 Y00), is referred to in the mediation analysis literature as a ’controlled
direct e¤ect’(CDE) of the exposure when …xing the mediator to levelM = 0. We can also consider controlled direct e¤ects which setM to a level other than0and in the Discussion section and further in the Appendix we consider four-way decompositions involving these alternative controlled direct e¤ects. The fourth component in the four-way decomposition, (Y01 Y00)(M1 M0), what we
referred to above as a ’mediated main e¤ect’is in fact equivalent to what in the mediation analysis literature is sometimes referred to as a ’pure indirect e¤ect’ (P IE). It is shown in the Appendix that:
P IE :=Y0M1 Y0M0 = (Y01 Y00)(M1 M0):
The counterfactual contrast, Y0M1 Y0M0, in the mediation analysis literature is referred to as a ’pure indirect e¤ect’1 or as a type of ’natural direct e¤ect’.2 This contrastY0M1 Y0M0 compares what would happen to the outcome if the mediator were changed from the level M0 (the level it
would be in the absence of the exposure) toM1 (the level it would be in the presence of exposure)
while in both counterfactual scenarios …xing the exposure itself to be absent. It will be non-zero if and only if the exposure changes the mediator (so thatM0 and M1 are di¤erent) and the mediator
itself has an e¤ect on the outcome even in the absence of the exposure. However, this is, in fact, the same quantity as what we had in our decomposition above, namely(Y01 Y00)(M1 M0). Note that
writing the pure indirect e¤ect as (Y01 Y00)(M1 M0)gives a representation of the pure indirect
e¤ect that does not require nested counterfactuals of the form Y0M1. This may be of interest as sometimes objections are made to the pure indirect e¤ect on the grounds that nested counterfactuals of the formY0M1 are di¢ cult to interpret. The third component,(Y11 Y10 Y01+Y00)(M1 M0), was recently considered in the mediation analysis literature and called a ’mediated interaction’ (IN Tmed).34 As discussed in the Appendix and elsewhere34, this mediated interaction can also be
written as (Y1M1 Y0M1 Y1M0 +Y0M0). The component we have not yet considered, the second component, (Y11 Y10 Y01+Y00)(M0), what we referred to above as a ’reference interaction’
(IN Tref) has no analogue in the current literature. However, it is shown in the Appendix that the
sum of the …rst and second component does have an analogue in the mediation analysis literature and it is equal to what is sometimes called in the mediation analysis literature the ’pure direct e¤ect’(P DE) de…ned as Y1M0 Y0M0 which compares what would happen to the outcome in the presence versus the absence of the exposure if, in both cases, the mediator were set to whatever it would be for that individual in the absence of exposure. In other words we have that
P DE :=Y1M0 Y0M0 = (Y10 Y00) + (Y11 Y10 Y01+Y00)(M0)
= CDE+IN Tref:
The pure direct e¤ect is the sum of a controlled direct e¤ect and, our second component, the reference interaction. If in our four-way decomposition above we replace the …rst two components with the pure direct e¤ect and write the fourth component as the pure indirect e¤ect we obtain:
Y1 Y0=P DE+IN Tmed+P IE: (3)
In other words, we can decompose the total e¤ect into a pure direct e¤ect, a pure indirect e¤ect, and a mediated interaction. This decomposition in (3) was the three-way decomposition provided by VanderWeele34 in 2013. However, even this three-way decomposition is relatively new and prior to this, a two-way decomposition was the norm in the mediation analysis literature. As discussed in the Appendix and in VanderWeele,34 the sum of the mediated interaction and the pure indirect e¤ect is equal to what in the mediation analysis literature is sometimes called a ’total indirect e¤ect’ (T IE), de…ned as Y1M1 Y1M0. Whereas, the pure indirect e¤ect, Y0M1 Y0M0, compares changing the mediator from M0 to M1 while …xing the exposure itself to be absent,
the total indirect e¤ect, Y1M1 Y1M0, compares changing the mediator from M0 to M1 …xing the exposure to present. With the total indirect so de…ned we have T IE = P IE +IN Tmed i.e.
(Y1M1 Y1M0) = (Y0M1 Y0M0) + (Y11 Y10 Y01+Y00)(M1 M0). We can then combine the
mediated interaction and the pure indirect e¤ect in the decomposition in (3), into a total indirect e¤ect to obtain the more standard 2-way decomposition in the mediation analysis literature:
Y1 Y0=P DE+T IE: (4)
This is the decomposition that has been used most often in the causal inference literature when assessing direct and indirect e¤ects; this two-way decomposition was …rst proposed in 1992 by Robins and Greenland1; and it is the decomposition that most of the existing software packages for
causal mediation analysis have focused on.8;16 This two-way decomposition also provides the coun-terfactual formalization for the decompositions typically employed in the social science literature on mediation.35 However, as we have seen above, the pure direct e¤ect is itself a combination of
two components: a controlled direct e¤ect and the reference interaction(P DE=CDE+IN Tref).
And the total indirect e¤ect is a combination of two components, the pure indirect e¤ect and the mediated interaction (T IE = P IE +IN Tmed). When these e¤ects are estimated on
aver-age, a proportion mediated measure, EE[T IE[T E]], is sometimes used which can also be re-written as
E[T IE]
E[T E] =
E[IN Tmed]+E[P IE]
E[T E] .
Yet another decomposition is worth noting in the mediation analysis literature. Sometimes the mediated interaction in the decomposition in (3) is combined with pure direct e¤ect, rather than with the pure indirect e¤ect, for an alternative two-way decomposition. As discussed in the Appendix and in VanderWeele34, the sum of the mediated interaction and the pure direct e¤ect is equal to what in the mediation analysis literature is sometimes called a ’total direct e¤ect’(T DE),1 de…ned as Y1M1 Y0M1. The total and the pure direct e¤ects are sometimes also called ’natural direct e¤ects’2 and the total and the pure indirect e¤ects are sometimes called ’natural indirect e¤ects’2. A summary of the various composite e¤ects is given in Table 4.
Table 4. Composite E¤ects
E¤ect Counterfactual De…nition Composite Relationship Total Indirect E¤ect (T IE) (Y1M1 Y1M0) T IE=P IE+IN Tmed
Pure Direct E¤ect (P DE) (Y1M0 Y0M0) P DE=CDE+IN Tref
Total Direct E¤ect (T DE) (Y1M1 Y0M1) T DE=CDE+IN Tref+IN Tmed
Portion Eliminated (P E) (Y1 Y0) (Y10 Y00) P E=P IE+IN Tref+IN Tmed Portion Attributable to Interaction (P AI) (Y11 Y10 Y01+Y00)(M1) P AI=IN Tref+IN Tmed
Of interest here is that the total direct e¤ect contains three components: the controlled direct e¤ect, the reference interaction, and the mediated interaction. As we move from the …rst to third of these components, we see they increasing involve the mediator in more substantial ways. The controlled direct e¤ect, (Y10 Y00), operates completely independent of the mediator; for this to
be non-zero the direct e¤ect must be present even when the mediator is absent. The reference interaction, (Y11 Y10 Y01+Y00)(M0), requires the mediator to operate but the e¤ect does not
come about by the exposure changing the mediator - it simply requires that the mediator itself is present even when the exposure is absent; the e¤ect is ’unmediated’ in the sense that it does not operate by the exposure changing the mediator, but it requires the presence of the mediator nonetheless. The third component, the mediated interaction, (Y11 Y10 Y01+Y00)(M1 M0),
is a type of mediated e¤ect; it requires that the exposure change the mediator; but it is also a direct e¤ect insofar as an interaction must also be present (the e¤ect of the exposure is di¤erent for di¤erent levels of the mediator); the third component thus not only involves the mediator but it is a mediated e¤ect, and a direct e¤ect as well. It is for this reason that it is sometimes combined with the pure indirect e¤ect to obtain the total indirect e¤ect, and sometimes combined with the pure direct e¤ect to obtain the total direct e¤ect.
When we combine the pure direct e¤ect and mediated interaction to get the total direct e¤ect, T DE:=Y1M1 Y0M1 =P DE+IN Tmed, we have the alternative 2-way decomposition of the total e¤ect into the sum of the total direct e¤ect and the pure indirect e¤ect:
Y1 Y0=T DE+P IE: (5)
This decomposition was likewise proposed by Robins and Greenland1 in 1992. Relatively easy-to-use software is currently available to estimate the components of the two-way decompositions in (4) and (5) on average for a population, under the assumptions described later in the paper. Note that in the decomposition in (5), the total direct e¤ect consists of three of the four basic components (the controlled direct e¤ect, the reference interaction, and the mediated interaction), whereas the
pure indirect e¤ect constitutes a single component. The mediated interaction is, however, arguably part of the e¤ect that is mediated and thus, when questions of mediation are of interest, it is arguably (4), rather than (5), that is to be preferred when assessing the extent of mediation.9;10;34 However, whether the pure indirect e¤ect or the total indirect e¤ect is of interest may depend upon the context.6
A …nal measure that is used in the mediation analysis literature is sometimes referred to as the "portion eliminated" (P E).1;12 As noted above, this is generally de…ned as the di¤erence between the total e¤ect and the controlled direct e¤ect: P E := (Y1 Y0) CDE. It is the portion of the
e¤ect of the exposure that would remain if the mediator were …xed to 0. The portion eliminated may be of interest insofar as it allows one to assess how much of the e¤ect of the exposure can be eliminated or prevented by intervening on the mediator; for this reason it is sometimes argued to be of policy interest.1;6;12 The four-way decomposition above in fact shows that this portion
eliminated measure is equal to the sum of the other three components: the reference interaction, the mediated interaction, and the pure indirect e¤ect i.e. P E = IN Tref +IN Tmed+P IE and
we can write the total e¤ect as T E=CDE+P E. The four-way decomposition provides a causal interpretation for the di¤erence between the total e¤ect and the controlled direct e¤ect: it is the portion of the e¤ect attributable to mediation, or interaction, or both. When the portion eliminated is estimated at the population level, sometimes a proportion eliminated measure is also calculated as
E[T E] E[CDE]
E[T E] which we could also rewrite as
E[IN Tref]+E[IN Tmed]+E[P IE]
E[T E] ; note that this is di¤erent
from the proportion mediated measure considered earlier which was EE[N IE[T E]] = E[IN Tmed]+E[P IE]
E[T E] .
The proportion eliminated includes in the numerator the reference interaction (since this part of the e¤ect is eliminated if the mediator is removed); the proportion mediated does not include the reference interaction in the numerator (since this is not part of the mediated e¤ect).12
We have seen then a number of di¤erent decompositions. However, when we are interested in questions of mediation, we need not choose between the two-way decompositions, or even the three-way decomposition, but can in fact use the decomposition into four components above so as to assess the portion of the total e¤ect that is attributable just to mediation, just to interaction, to both mediation and interaction, or to neither mediation nor interaction. The four-way decomposition allows us to accomplish this. The various decompositions within the context of mediation are summarized in Table 5, but the four-way decomposition here essentially provides a framework which encompasses them all.
Table 5. Mediation Decompositions Number of Components Decomposition 2-Way Decompositiona T E=T IE+P DE 2-Way Decompositionb T E=T DE+P IE 2-Way Decompositionc T E=CDE+P E
3-Way Decompositiond T E=P DE+P IE+IN Tmed
4-Way Decompositione T E=CDE+IN Tref +IN Tmed+P IE
a(Y1 Y0) = (Y1M1 Y1M0) + (Y1M0 Y0M0)
b(Y1 Y0) = (Y1M1 Y0M1) + (Y0M1 Y0M0)
c(Y1 Y0) = (Y10 Y00) + [(Y1 Y0) (Y10 Y00)]
d(Y1 Y0) = (Y0M1 Y0M0) + (Y1M0 Y0M0) + (Y11 Y10 Y01+Y00)(M1 M0)
Relation to Interaction Decompositions
VanderWeele and Tchetgen Tchetgen24 recently considered attributing a portion of the total e¤ect of one exposure on an outcome that is due to an interaction with a second exposure. Here we will relate this to the four-way decomposition above. Our four-way decomposition above was expressed as:
Y1 Y0 = (Y10 Y00) + (Y11 Y10 Y01+Y00)(M0) (1)
+(Y11 Y10 Y01+Y00)(M1 M0) + (Y01 Y00)(M1 M0):
which we also wrote as: T E =CDE +IN Tref +IN Tmed+P IE. Suppose now that instead of
considering how much of the total e¤ect is mediated versus direct, as in the previous section, we were interested in the portion due to interaction. In our four-way decomposition, two of the four components (the second and the third involve) an interaction. We could thus de…ne the portion due to interaction as their sum: P AI :=IN Tref+IN Tmed= (Y11 Y10 Y01+Y00)(M0) + (Y11
Y10 Y01+Y00)(M1 M0) = (Y11 Y10 Y01+Y00)(M1) and we would then have the 3-way
decomposition:
T E = CDE+P AI+P IE (6)
= (Y10 Y00) + (Y11 Y10 Y01+Y00)(M1) + (Y01 Y00)(M1 M0):
The total e¤ect can be decomposed into the e¤ect of A with M absent (CDE), a pure indirect e¤ect (P IE), and a portion due to interaction (P AI). Consider now the empirical analogue of this decomposition using the expressions in (1b). We let pam = E[YjA = a; M = m] and, pa =
E[YjA=a], andpm=E[YjM =m]and we have from (1b): (pa=1 pa=0) =
(p10 p00)+(p11 p10 p01+p00)P(M = 1jA= 1)+(p01 p00)fP(M = 1jA= 1) P(M = 1jA= 0)g:
(7) We again have the decomposition of the average total e¤ect ofA, into what is essentially the average controlled direct e¤ect, the average portion attributable to interaction, and the average pure indirect e¤ect. The middle component is the component due to interaction and the proportion of the e¤ect due to interaction could then be assessed by: (p11 p10 p01+p00)P(M = 1jA= 1)=(pa=1 pa=0).
In fact, the decomposition given above in (7) is that which VanderWeele and Tchetgen24 used
when attributing e¤ects to interactions. Several points are worth noting. First, the decomposition in (6) and (7) for the portion attributable to interaction follows quite clearly from the four-way decomposition. The decomposition in (6) is the decomposition at the individual counterfactual level analogous to the empirical decomposition in (7) given by VanderWeele and Tchetgen Tchetgen.24 Second, VanderWeele and Tchetgen Tchetgen considered two cases, one in which A and M were independent and one in which they are not. The decomposition in (7) was that which was proposed whenA a¤ectedM. WhenAand M are independent, the decomposition in (7) reduces to(pa=1
pa=0) = (p10 p00)+(p11 p10 p01+p00)P(M = 1)and we likewise have a similar decomposition for
the total e¤ect ofM onY: (pm=1 pm=0) = (p01 p00) + (p11 p10 p01+p00)P(A= 1). Likewise,
on a ratio scale, when A does not a¤ect M, the third and fourth components in Table 2 become
0and we are left with P ACDE = (RR10 1)+((RRRERI10 1))P(M=1) and P AIN T ref = (RR10(RERI1)+(RERI)P(M=1))P(M=1)
which are also the expressions given by VanderWeele and Tchetgen Tchetgen24 for attributing e¤ects to interactions on a ratio scale. When Aa¤ectsM, the decomposition for the total e¤ect of A on Y is altered and we must use the decomposition in (7). Finally, when A does not a¤ectY, we have an analogous individual counterfactual level decomposition as (6) then reduces to: T E=
(Y10 Y00) + (Y11 Y10 Y01+Y00)(M)since whenAdoes not a¤ectM,M1 =M0=M. All of this
also follows from our four-way decomposition which encompasses all of the prior decompositions. These decompositions are all summarized in Table 6.
Table 6. Interaction Decompositions
Number of Components Decomposition
2-Way Decomposition (No Mediation)a T E =CDE+P AI
3-Way Decompositionb T E =CDE+P AI+P IE
4-Way Decompositionc T E =CDE+IN Tref +IN Tmed+P IE
a(Y1 Y0) = (Y10 Y00) + (Y11 Y10 Y01+Y00)(M)
b(Y1 Y0) = (Y10 Y00) + (Y11 Y10 Y01+Y00)(M1) + (Y01 Y00)(M1 M0)
c(Y1 Y0) = (Y10 Y00) + (Y11 Y10 Y01+Y00)(M0) + (Y11 Y10 Y01+Y00)(M1 M0) + (Y01 Y00)(M1 M0)
Although in the more general setting when A a¤ects M, we can estimate the portion due to interaction on the average level using the three-way decomposition in (7), there is no need to use only a three-way decomposition; we can instead use the four-way decomposition in (1) and the empirical expressions in (1b) to further divide the portion due to interaction into that which is due to interaction but not mediation (the reference interaction,E[IN Tref]) and the portion due to
interaction and mediation (the mediated interactionE[IN Tmed]). Such a four-way decomposition,
in which the portion attributed to interaction is itself further divided may shed additional insight. Perhaps most importantly, this four-way decomposition, which helps better understand the portions of a total e¤ect due to interaction, is exactly the same decomposition that was used above to shed insight into what portions of the total e¤ect were mediated and which portions were direct. The same four-way decomposition was useful in assessing both mediation and interaction. The same four components are used in assessing mediation and interaction, but the components are combined in di¤erent ways to assess these di¤erent phenomena. However, the four-way decomposition itself essentially provides a uni…cation of these phenomena of mediation and interaction. The four-fold decomposition underlies the various more speci…c decompositions in assessing both mediation and interaction.
Figure 3: The four-fold decomposition encompasses both decompositions for mediation and interaction. For interaction, the reference interaction (IN Tref) and the mediated interaction
(IN Tmed) combine to the portion attributable to interaction(P AI). The portion attributable to
interaction(P AI) combine with the controlled direct e¤ect(CDE) and the pure indirect e¤ect
(P IE) to give the total e¤ect(T E). For mediation, the controlled direct e¤ect and the reference interaction(IN Tref)combine to give the pure direct e¤ect (P DE); the pure indirect e¤ect(P IE)
combines with the mediated interaction(IN Tmed) to give the total indirect e¤ect(T IE); and the
pure direct e¤ect (P DE) combines with total indirect e¤ect(T IE) to give the total e¤ect(T E).
As illustrated in Figure 3, the four components form the backbone of both the various mediation decompositions (Figures 3-5) and the interaction decomposition (Figure 3).
Figure 4: As an alternative mediation decomposition, the controlled direct e¤ect and the reference interaction(IN Tref)combine to give the pure direct e¤ect (P DE); the pure direct e¤ect(P DE)
and the mediated interaction(IN Tmed) combine to give the total direct e¤ect (T DE); and the
Figure 5: As an alternative mediation decomposition, the di¤erence between the total e¤ect(T E)
and the controlled direct e¤ect(P E)is sometimes called the portion elimianted (P E) and it is equal to the sum of the reference interaction (IN Tref), the mediated interaction(IN Tmed), and
the pure indirect e¤ect (P IE).
Once, again, however, the greatest insight is arguably gained when the four-fold approach is used to assess simultaneously the portions of the total e¤ect that are due just to mediation, just to interaction, to both mediation and interaction, and to neither mediation nor interaction.
Discussion
The four-way decomposition here encompasses and unites previous decompositions in the liter-ature, both concerning mediation and concerning interaction. The results here have also provided a mechanistic interpretation to the di¤erence between a total e¤ect and a controlled direct e¤ect; this contrast has been used to assess policy implications and it is more easily identi…ed than many other causal quantities concerning mediation; the results here show that it also has a mechanistic interpretation as well. We have also shown how the four-way decomposition in this paper can be carried out on a di¤erence scale and on a ratio scale, we have related the various components to standard statistical models, and in the eAppendix we have provided software code to carry out the estimation of the various components of the decomposition using such regression models. We have seen that in addition to reporting the four components, an investigator can also easily report, along with these, the overall proportion attributable to interaction, the overall proportion mediated, and the proportion of the e¤ect that would be eliminated if the mediator were removed. As seen in the empirical example in genetic epidemiology, the approach described here can shed considerable insight into the relationships between an exposure and a mediator with an outcome, and into the role of both mediation and interaction in these relationships.
In the text here we have focused on a binary exposure and binary mediator, and the controlled direct e¤ect we have been considering is that in which the mediator is …xed to being absent. Much more general results are given in the Appendix and the approach in fact applies to arbitrary
mediator to be absent, one can consider controlled direct e¤ects that …x the mediator to some other level, m . Similar four-way decompositions can be carried out wherein the …rst component is the controlled direct e¤ect with the mediator …xed to levelm . When this is done, the reference interaction term changes because, with the mediator …xed to m (rather than 0), the controlled direct e¤ect then picks up some of the e¤ect of the interaction between the exposure and the mediator. With the controlled direct e¤ect in which the mediator is …xed tom , the interpretation of the reference interaction is then the portion of the e¤ect due to the interaction between the exposure and the mediator that is not mediated, and also not captured by the controlled direct e¤ect. Again, the results in the Appendix cover very general settings and will thus likely be of use in a variety of contexts. The code in the eAppendix likewise provides practical and relatively easy-to-use software tools to implement the approaches here in a wide range of settings. The central limitations of the approach developed here is the strong assumptions being made about confounding; these are, however, similar assumptions to those made in the literature on mediation that only focuses on simpler decompositions. Future research could examine the robustness of each of the four components to confounding and measurement error. For example, recent work indicates that interaction terms may be more robust to confounding,36 but that interaction terms when the two exposures are correlated may be particularly sensitive to measurement error;37;38 di¤erent components may be robust to di¤erent forms of bias. Future work could also extend existing sensitivity analysis techniques for mediation and interaction7;8;36 38 to each of the four components.
Prior work on mediation within the counterfactual framework has accommodated potential interaction. The approach here makes the role of interaction, and its separate contribution be-yond mediation, clearer, and unites, within a single framework, the phenomena of mediation and interaction.
Appendix
In the Appendix we will no longer restrict attention to binary exposure and mediator and will consider an arbitrary exposure and mediator. We will assume we are comparing two exposure levels aand a . We give the general four-way decomposition result in Proposition 1.
Proposition 1. For any levelm of M we have Ya Ya
= (Yam Ya m ) +
X
m(Yam Ya m Yam +Ya m )1(Ma =m)
+X
m(Yam Ya m)f1(Ma=m) 1(Ma =m)g+ (Ya Ma Ya Ma )
Proof. We have that Ya Ya
= YaMa Ya Ma = (YaMa Ya Ma) + (Ya Ma Ya Ma ) = (YaMa Ya Ma ) + (Ya Ma Ya Ma ) + (YaMa Ya Ma YaMa +Ya Ma ) = (Yam Ya m ) +f(YaMa Ya Ma ) (Yam Ya m )g +(YaMa Ya Ma YaMa +Ya Ma ) + (Ya Ma Ya Ma ) = (Yam Ya m ) + X mf(Yam Ya m) (Yam Ya m )g1(Ma =m) +X mf(Yam Ya m)1(Ma =m) (Yam Ya m)1(Ma =m)g+ (Ya Ma Ya Ma ) = (Yam Ya m ) + X m(Yam Ya m Yam +Ya m )1(Ma =m) +X m(Yam Ya m)f1(Ma =m) 1(Ma =m)g+ (Ya Ma Ya Ma ):
The four components of the decomposition in general form are thus
CDE(m ) : = (Yam Ya m ) IN Tref(m ) : = X m(Yam Ya m Yam +Ya m )1(Ma =m) IN Tmed : = X m(Yam Ya m)f1(Ma=m) 1(Ma =m)g P IE : = (Ya Ma Ya Ma ):
Note we can also rewriteIN Tmed=
X
m(Yam Ya m Yam +Ya m )f1(Ma=m) 1(Ma =m)g
and we can rewriteP IE =X
m(Ya m Ya m )f1(Ma=m) 1(Ma =m)g. Doing so with binary
A and M and settinga= 1; a = 0; m = 0 gives us the decomposition in (1) in the text:
Y1 Y0 = (Y10 Y00) + (Y11 Y10 Y01+Y00)(M0) (1)
+(Y11 Y10 Y01+Y00)(M1 M0) + (Y01 Y00)(M1 M0):
Proposition 2. For any levelm of M we have E[Yja; c] E[Yja ; c] = = fE[Yja; m ; c] E[Yja ; m ; c]g + Z fE[Yja; m; c] E[Yja ; m; c] E[Yja; m ; c] +E[Yja ; m ; c]gdP(mja ; c) + Z fE[Yja; m; c] E[Yja ; m; c]gfdP(mja; c) dP(mja ; c)g + Z E[Yja ; m; c]fdP(mja; c) dP(mja ; c)g:
Proof. We have that E[Yja; c] E[Yja ; c]
= E[Yja; m ; c] E[Yja ; m ; c] +fE[Yja; c] E[Yja; m ; c]g fE[Yja ; c] E[Yja ; m ; c]g = E[Yja; m ; c] E[Yja ; m ; c] + Z fE[Yja; m; c] E[Yja; m ; c]gdP(mja; c) Z fE[Yja ; m; c] E[Yja ; m ; c]gdP(mja ; c) = fE[Yja; m ; c] E[Yja ; m ; c]g + Z fE[Yja; m; c] E[Yja ; m; c]g fE[Yja; m ; c] E[Yja ; m ; c]gdP(mja; c) + Z fE[Yja ; m; c] E[Yja ; m ; c]gfdP(mja; c) dP(mja ; c)g = fE[Yja; m ; c] E[Yja ; m ; c]g + Z fE[Yja; m; c] E[Yja ; m; c]g fE[Yja; m ; c] E[Yja ; m ; c]gdP(mja ; c) + Z fE[Yja; m; c] E[Yja ; m; c]g fE[Yja; m ; c] E[Yja ; m ; c]gfdP(mja; c) dP(mja ; c)g + Z fE[Yja ; m; c] E[Yja ; m ; c]gfdP(mja; c) dP(mja ; c)g: = fE[Yja; m ; c] E[Yja ; m ; c]g + Z fE[Yja; m; c] E[Yja ; m; c] E[Yja; m ; c] +E[Yja ; m ; c]gdP(mja ; c) + Z fE[Yja; m; c] E[Yja ; m; c]gfdP(mja; c) dP(mja ; c)g + Z E[Yja ; m; c]fdP(mja; c) dP(mja ; c)g:
Note we can also rewrite the third term as
Z
fE[Yja; m; c] E[Yja ; m; c]g fE[Yja; m ; c]
E[Yja ; m ; c]gfdP(mja; c) dP(mja ; c)g and the fourth term as
Z
and setting a= 1; a = 0; m = 0 gives decomposition (1b) in the text:
pa=1 pa=0 = (p10 p00) + (p11 p10 p01+p00)P(M = 1jA= 0)
+ (p11 p10 p01+p00)fP(M = 1jA= 1) P(M = 1jA= 0)g
+ (p01 p00)fP(M = 1jA= 1) P(M = 1jA= 0)g (1b)
Note the decomposition in Proposition 2 is a property of the expectations and probabilities. It does not require confounding assumptions. However, to interpret the components as causal e¤ects, confounding assumptions are required. We will begin our discussion of confounding by …rst considering non-parametric structural equations.18 Consider the following four confounding assumptions: (i) the e¤ect the exposure A on the outcome Y is unconfounded conditional on C; (ii) the e¤ect the mediator M on the outcome Y is unconfounded conditional on (C; A); (iii) the e¤ect the exposure A on the mediator M is unconfounded conditional on C; and (iv) none of the mediator-outcome confounders are themselves a¤ected by the exposure. If we letX??YjZ denote thatX is independent ofY conditional onZ, then these four assumptions stated formally in terms of counterfactual independence are: (i) Yam ??AjC, (ii) Yam ??MjfA; Cg, (iii) Ma ??AjC, and
(iv)Yam??Ma jC.
Proposition 3. Under assumptions (i)-(iv) we have: E[CDE(m )jc] = fE[Yja; m ; c] E[Yja ; m ; c]g E[IN Tref(m )jc] = Z fE[Yja; m; c] E[Yja ; m; c] E[Yja; m ; c] +E[Yja ; m ; c]dP(mja ; c) E[IN Tmedjc] = Z fE[Yja; m; c] E[Yja ; m; c]gfdP(mja; c) dP(mja ; c)g E[P IEjc] = Z E[Yja ; m; c]fdP(mja; c) dP(mja ; c)g:
Proof. The …rst equality is established by Robins39, the fourth by Pearl2, the third by VanderWeele34. For the second equality we haveE[IN Tref(m )jc]
= EhX m(Yam Ya m Yam +Ya m )1(Ma =m)jc i = Z m E[Yam Ya m Yam +Ya m jc]dP(Ma =mjc) = Z mf E[Yamjc] E[Ya mjc] E[Yam jc] +E[Ya m jc]gdP(Ma =mjc) = Z mf E[Yamja; m; c] E[Ya mja ; m; c] E[Yam ja; m ; c] +E[Ya m ja ; m ; c]gdP(Ma =mja ; c) = Z mf E[Yja; m; c] E[Yja ; m; c] E[Yja; m ; c] +E[Yja ; m ; c]gdP(M =mja ; c)
where the second equality follows by assumption (iv) and the fourth by assumptions (i)-(iii). In fact, the other three equalities in Proposition 3 can be established in much the same way.
We can also interpret the terms in the decomposition in Proposition 2 causally under assump-tions (i)-(iii) alone, though the causal interpretation is slightly weaker.
Proposition 4. Under assumptions (i)-(iii) we have: fE[Yja; m ; c] E[Yja ; m ; c]g=E[Yam Ya m jc] Z fE[Yja; m; c] E[Yja ; m; c] E[Yja; m ; c] +E[Yja ; m ; c]gdP(mja ; c) = Z E[Yam Ya m Yam +Ya m jc]dP(Ma jc) Z fE[Yja; m; c] E[Yja ; m; c]gfdP(mja; c) dP(mja ; c)g = Z E[Yam Ya mjc]fdP(Majc) dP(Ma jc) Z E[Yja ; m; c]fdP(mja; c) dP(mja ; c)g= Z E[Ya mjc]fdP(Majc) dP(Ma jc)g: :
Proof. The …rst equality is established by Robins39, the second in the …nal four lines of the
proof of Proportion 3 above, the third in VanderWeele34 and the fourth, using slightly di¤erent notation by Didelez et al.40
Note we can also rewrite the right hand side of the third equality as
Z
fE[Yam Ya mjc]
E[Yam Ya m jc]gfdP(Majc) dP(Ma jc) and the right hand side of the fourth equality as
Z
fE[Ya m Ya m jc]gfdP(Majc) dP(Ma jc)g. The right hand side of the equalities in Proposition
4 are causal quantities but rather than directly taking population averages of the four components of the decomposition, the e¤ects of A and M on Y are integrated over the distribution of M under di¤erent exposure settings. As discussed further in the eAppendix, these e¤ects can be inter-preted as randomized interventional analogues of the four components of the decomposition. They only require assumptions (i)-(iii) for identi…cation (i.e. they do not require the more controver-sial cross-world independence assumption (iv)) but the causal interpretation of these randomized interventional analogues is somewhat weaker.
Finally, we note that some earlier literature on interaction with binary exposures made use of di¤erent response types where individuals were classi…ed according to their joint counterfactual outcomes(Y00; Y01; Y10; Y11). Under the response type classi…cation for mediation given by Hafeman
and VanderWeele41 in which it is assumed that the monotonicity assumption that Y
am is
non-decreasing in a and in m holds, the four components of the four-decomposition could be written
asCDE(0) = Y(2)+ Y(4),IN Tref(0) =M(1)(Y(8) Y(2)),IN Tmed=M(2)(Y(8) Y(2)), and P IE =
M(2)(Y(2)+Y(6)) whereM(1) is a binary indicator such thatM(1) = 1ifM0 =M1 = 1andM(1) = 0
otherwise;M(2)is a binary indicator such thatM(2) = 1ifM0= 0; M1 = 1andM(2) = 0otherwise;
Y(2) is a binary indicator such that Y(2) = 1 if (Y00 = 0; Y01 = 1; Y10 = 1; Y11 = 1) and Y(2) = 0
otherwise; Y(4) is a binary indicator such that Y(4) = 1 if(Y00 = 0; Y01 = 0; Y10 = 1; Y11 = 1) and
Y(4) = 0otherwise;Y(6)is a binary indicator such thatY(6) = 1if(Y00= 0; Y01= 1; Y10= 0; Y11= 1)
and Y(6) = 0 otherwise; andY(8) is a binary indicator such thatY(8) = 1 if(Y00= 0; Y01= 1; Y10= 1; Y11= 1)and Y(8)= 0 otherwise.
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eAppendix for "A uni…cation of mediation and interaction: a four-way decomposition" by Tyler J. VanderWeele
1. Continuous Outcomes and Linear Regression Models
1.1 Continuous Outcome, Continuous Mediator
For Y andM continuous, under assumptions (i)-(iv) and correct speci…cation of the regression models forY and M:
E[Yja; m; c] = 0+ 1a+ 2m+ 3am+ 04c
E[Mja; c] = 0+ 1a+ 02c;
VanderWeele and Vansteelandt4 and VanderWeele34 showed that the average controlled direct ef-fect, the pure indirect e¤ect, and the mediated interaction conditional on covariates C = c were given by:
E[CDE(m )jc] = ( 1+ 3m )(a a )
E[P IEjc] = ( 2 1+ 3 1a )(a a )
E[IN Tmedjc] = 3 1(a a )(a a ):
They also showed that the pure direct e¤ect was given by E[P DEjc] = f 1 + 3( 0 + 1a +
0
2c)g(a a ). The reference interaction is then given by di¤erence between the the pure direct
e¤ect and the controlled direct e¤ect:
E[IN Tref(m )jc] = f 1+ 3( 0+ 1a + 02c)g(a a ) ( 1+ 3m )(a a )
= 3( 0+ 1a + 20c m )g(a a ):
Standard errors for these expressions could be derived using the delta method along the lines of the derivations in VanderWeele and Vansteelandt4 or by using bootstrapping.
1.2 Continuous Outcome, Binary Mediator
For Y continuous and M binary, under assumptions (i)-(iv) and correct speci…cation of the regression models for Y and M:
E[Yja; m; c] = 0+ 1a+ 2m+ 3am+ 04c
logitfP(M = 1ja; c)g = 0+ 1a+ 02c:
Valeri and VanderWeele16 show that the average controlled direct e¤ect and the average pure indirect e¤ect are given by:
E[CDE(m )jc] = ( 1+ 3m )(a a ) E[P IEjc] = ( 2+ 3a )f exp[ 0+ 1a+ 02c] 1 +exp[ 0+ 1a+ 02c] exp[ 0+ 1a + 02c] 1 +exp[ 0+ 1a + 02c]g:
The reference interaction is given by the di¤erence between the pure direct e¤ect and the controlled direct e¤ect, which were both given by Valeri and VanderWeele16:
E[IN Tref(m )jc] = f 1(a a )g+f 3(a a )g exp[ 0+ 1a + 02c] 1 +exp[ 0+ 1a + 02c] ( 1+ 3m )(a a ) = 3(a a ) exp[ 0+ 1a + 02c] 1 +exp[ 0+ 1a + 02c] m !
The mediated interaction is given by the di¤erence between the total indirect e¤ect and the pure indirect e¤ect, which were also both given by Valeri and VanderWeele16:
E[IN Tmedjc] = ( 2+ 3a)f exp[ 0+ 1a+ 02c] 1 +exp[ 0+ 1a+ 02c] exp[ 0+ 1a + 02c] 1 +exp[ 0+ 1a + 02c]g: ( 2+ 3a )f exp[ 0+ 1a+ 02c] 1 +exp[ 0+ 1a+ 02c] exp[ 0+ 1a + 02c] 1 +exp[ 0+ 1a + 02c]g = 3(a a )f exp[ 0+ 1a+ 0 2c] 1 +exp[ 0+ 1a+ 02c] exp[ 0+ 1a + 02c] 1 +exp[ 0+ 1a + 02c]g:
2. Decomposition on a Ratio Scale and Logistic Regression Models
2.1. Four-way Decomposition on a Ratio Scale
From Proposition 1 in the text we have Ya Ya
= (Yam Ya m ) +
X
m(Yam Ya m Yam +Ya m )1(Ma =m)
+X
m(Yam Ya m)f1(Ma=m) 1(Ma =m)g+ (Ya Ma Ya Ma ):
Taking expectations conditional onC =c gives: E(Ya Ya jc)
= E(Yam Ya m jc) +
X
mE[(Yam Ya m Yam +Ya m )1(Ma =m)jc]
+X
mE[(Yam Ya m)f1(Ma=m) 1(Ma =m)gjc] +E(Ya Ma Ya Ma jc):
Under assumption (iv) this is: E(Ya Ya jc)
= E(Yam Ya m jc) +
X
mE(Yam Ya m Yam +Ya m jc)P(Ma =mjc)
+X
mE(Yam Ya mjc)fP(Ma=mjc) P(Ma =mjc)g+E(Ya Ma Ya Ma jc):
and dividing by E(Ya jc) gives:
whereRRT E c = E(Yajc) E(Ya jc), = E(Ya m jc) E(Ya jc) , and RRCDEc (m ) = E(Yam jc) E(Ya m jc) RRIN Tc ref(m ) = X mRERI(a ; m )P(Ma =mjc) RRIN Tmed c = X mRERI(a ; m )fP(Ma=mjc) P(Ma =mjc)g RRP IEc = E(Ya Majc) E(Ya Ma jc) with RERI(a ; m ) = E(Yamjc) E(Ya m jc) E(Ya mjc) E(Ya m jc) E(Yam jc)
E(Ya m jc)+ 1 . Under assumptions (i)-(iii) we
also have E(Yajc) = E(Yja; c), E(Yamjc) =
X
mE[Yja; m; c]P(mja; c) and thus and P(Ma =
mjc) = P(M = mja; c) and thus the right hand side of the equalities above would be identi…ed from the data. VanderWeele34 also showed that RRIN Tmed
c = X mRERI(a ; m )fP(Ma = mjc) P(Ma = mjc)g = EE[Y[YaMajc] a Ma jc] E[YaMa jc] E[Ya Ma jc] E[Ya Majc]
E[Ya Ma jc]+ 1 and called this latter term
RERImediated.
Note also under assumption (iv), (RRP IE
c 1)can be rewritten as (RRP IEc 1) = E(Ya Majc) E(Ya jc) E(Ya jc) E(Ya jc) = E(Ya m jc)f E(Ya Majc) E(Ya jc)g = E(Ya m jc) X mfE[Ya mjc] E[Ya m jc]gfP(Ma=mjc) P(Ma =mjc)g = X m E(Ya mjc) E(Ya m jc) 1 fP(Ma=mjc) P(Ma =mjc)g = X m E(Ya mjc) E(Ya m jc)f P(Ma=mjc) P(Ma =mjc)g
The proportion attributable to each of the four components is then obtained by simply dividing each of the four components in the display equation above by their sum as in Table 2. A similar decomposition could likewise be carried out on an additive scale using hazard ratios.
By similar arguments to those above but applied to Propositions 2 and 4, if assumption (iv) did not hold but assumptions (i)-(iii) all did hold, we would have that (RRT Ec 1) decomposed into
